SOME REMARKS ON TRANSVERSALLY HARMONIC MAPS
Glasgow mathematical journal, Tome 50 (2008) no. 1, pp. 1-16

Voir la notice de l'article provenant de la source Cambridge University Press

We consider transversally harmonic foliated maps between two Riemannian manifolds equipped with Riemannian foliations. We give various characterisations of such maps and we study the relation between the properties ‘harmonic’ and ‘transversally harmonic’ for a given map. We also consider these problems for particular classes of manifolds: manifolds with transversally almost Hermitian foliations and Riemannian flows.
DOI : 10.1017/S001708950700403X
Mots-clés : 53C12, 58E20
KONDERAK, JERZY J.; WOLAK, ROBERT. SOME REMARKS ON TRANSVERSALLY HARMONIC MAPS. Glasgow mathematical journal, Tome 50 (2008) no. 1, pp. 1-16. doi: 10.1017/S001708950700403X
@article{10_1017_S001708950700403X,
     author = {KONDERAK, JERZY J. and WOLAK, ROBERT},
     title = {SOME {REMARKS} {ON} {TRANSVERSALLY} {HARMONIC} {MAPS}},
     journal = {Glasgow mathematical journal},
     pages = {1--16},
     year = {2008},
     volume = {50},
     number = {1},
     doi = {10.1017/S001708950700403X},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S001708950700403X/}
}
TY  - JOUR
AU  - KONDERAK, JERZY J.
AU  - WOLAK, ROBERT
TI  - SOME REMARKS ON TRANSVERSALLY HARMONIC MAPS
JO  - Glasgow mathematical journal
PY  - 2008
SP  - 1
EP  - 16
VL  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S001708950700403X/
DO  - 10.1017/S001708950700403X
ID  - 10_1017_S001708950700403X
ER  - 
%0 Journal Article
%A KONDERAK, JERZY J.
%A WOLAK, ROBERT
%T SOME REMARKS ON TRANSVERSALLY HARMONIC MAPS
%J Glasgow mathematical journal
%D 2008
%P 1-16
%V 50
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S001708950700403X/
%R 10.1017/S001708950700403X
%F 10_1017_S001708950700403X

[1] 1.Baird, P. and Wood, J. C., Harmonic morphisms between Riemannian manifolds (Oxford Univesity Press, 2003). Google Scholar | DOI

[2] 2.Barletta, E. and Dragomir, S., On transversally holomorphic maps of Kählerian foliations, Acta Appl. Math. 54 (1998), 121–134. Google Scholar | DOI

[3] 3.Bierstone, E., The structure of orbit spaces and the singularities of equivariant mappings, Mon. de Mat. 35 (IMPA, Rio de Janeiro, 1980). Google Scholar

[4] 4.Black, M.: Harmonic maps into homogenous spaces, Pitman research notes in math., Longman Scientific and Technical, Harlow, 1991 Google Scholar

[5] 5.Cairns, G., A general description of totally geodesic foliations, That o huku Math. J. 38 (1986), 37–55. Google Scholar

[6] 6.Cairns, G. and Ghys, E., Totally geodesic foliations on 4-manifolds, J. Diff. Geom. 23 (1986), 241–254. Google Scholar

[7] 7.Blair, D.E.: Riemannian geometry of contact and sym-plec-tic ma-ni-folds, Birk-häu-ser, Basel, 2001 Google Scholar | DOI

[8] 8.Boualem, H., Molino, P.: Modèle locaux saturés de feuilletages riemanniens singuliers, C. R. Acad. Sci. Paris 316 (1993), 913–916 Google Scholar

[9] 9.Cannas, A. da Silva: Symplectic Geometry, LNM 1764, Springer 2001 Google Scholar

[10] 10.Cordero, L. A., Wolak, R.: Examples of foliations with foliated geometric structures, Pacific J. Math. 142, 2 1990, 265-276 Google Scholar | DOI

[11] 11.Dominguez, D., Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), 1237–1276. Google Scholar | DOI

[12] 12.Dombrowski, P., On the geometry of the tangent bundles, J. Reine Angew. Math. 210 (1962), 73–88. Google Scholar | DOI

[13] 13.Edwards, R., Millett, K. and Sullivan, D., Foliations with all leaves compact, Topology 16 (1977), 13–32. Google Scholar | DOI

[14] 14.Eells, J. and Fuglede, B., Harmonic maps between Riemannian polyhedra (Cambridge University Press, 2001). Google Scholar

[15] 15.Eells, J. and Lemaire, L., A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68. Google Scholar | DOI

[16] 16.Eells, J. and Lemaire, L., Selected topics in harmonic maps, C.B.M.S., Regional Conference Series, 57 (AMS., Providence, R.I. 1983). Google Scholar | DOI

[17] 17.Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524. Google Scholar | DOI

[18] 18.Eells, J., Verjovsky, A.: Harmonica and Riemannian foliations, Bol. Soc. Mat. Mexicana (3) (1998), 1–12 Google Scholar

[19] 19.Eells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. Google Scholar | DOI

[20] 20.Goresky, M. and MacPherson, R., Intersection homology theory, Topology 19 (1980), 135–165. Google Scholar | DOI

[21] 21.Kacimi–Alaoui, A. El: Opèrateur transversalement elliptic sur un feuilletage riemannien et applications, Comp. Math. 73, (1990), 57–106 Google Scholar

[22] 22.Fujimoto, A.: Theory of G–structures, Tokyo, 1972 Google Scholar

[23] 23.Gauduchon, P.: Alcuni spunti di geometria quasi hermitiana e hermitiana, Quaderni del seminario di Top. Algerbrica e Diff., Univ. Rome, 1983 Google Scholar

[24] 24.Ghys, E., Classification des feuilletages totalement géodésiques de codimension un, Comm. Math. Helv. 58 (1983), 543–572. Google Scholar | DOI

[25] 25.Gray, A. and Hervella, L., The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali Mat. Pura Appl. 123 (1980), 35–58. Google Scholar | DOI

[26] 26.Helgason, S., Differential geometry of symmetric spaces (Academic Press, 1962). Google Scholar

[27] 27.Ianuš, S.: Sulle strutture canoniche dello spazio fibrato tangente di una varietà riemanniana, Rendiconti di Matematica, Univ. di Roma, VI, 6, fasc.1 (1973), 75–96 Google Scholar

[28] 28.Hermann, R., A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle, Proc. AMS 11 (1960), 236–242 Google Scholar

[29] 29.Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215–229. Google Scholar

[30] 30.Józefowicz, M. and Wolak, R., A few remarks on the geometry of the space of leaf closures of a Riemannian foliation, in Proceedings of a year in Differential Geometry University of Maryland (Banach Center Publications), Vol. 76 (2007), 395–409. Google Scholar

[31] 31.Kamber, F. and Tondeur, Ph., Foliations and metrics, Proceedings of a Year in Differential Geometry, University of Maryland (Birkhäuser, 1983), 103–152 Google Scholar

[32] 32.Kamber, F. and Tondeur, Ph., Duality theorems for foliations, Astérisque 116 (1984), 108–116. Google Scholar

[33] 33.Kobayashi, S., Transformation groups in differential geometry (Springer, Verlag, 1972). Google Scholar | DOI

[34] 34.Konderak, J.J., On natural first order Lagrangians, Bull. London Math. Soc. 23 (1991), 169–174 Google Scholar | DOI

[35] 35.Konderak, J. and Wolak, R., On transversally harmonic maps between manifolds with Riemannian foliations, Quart. J. Math. Oxford Ser. (2) 54 (2003), 335–354. Google Scholar | DOI

[36] 36.Kowalski, O., Curvature of the Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine und Angew. Math. 220 (1971), 124–129. Google Scholar

[37] 37.Lichnerowicz, A., Applications harmoniques et variétés kähleriennes, Sympos. Math. 3 Bologna (1970), 341–402. Google Scholar

[38] 38.Masa, X., Duality and minimality in Riemannian foliations, Comm. Math. Helv. 67 (1992), 17–27. Google Scholar | DOI

[39] 39.Miquel, V. and Wolak, R., Minimal singular Riemannian foliations, CRAS 342 (2006), 33–37. Google Scholar

[40] 40.Molino, P., Riemannian foliations (Birkhäuser, 1988). Google Scholar | DOI

[41] 41.Molino, P.: Orbit–like foliations, Geometric Study of Foliations, Proc. Tokyo 1993, World Scientific, Singapore, (1994), 97–119 Google Scholar

[42] 42.Moore, C. and Schochet, C., Global analysis on foliated spaces, (Springer-Verlag, 1988). Google Scholar | DOI

[43] 43.Mostow, M., Continuous cohomology of spaces with two topologies, Mem. Amer Math. Soc. 7 (1976), no. 175. Google Scholar

[44] 44.Pierrot, M.: Orbites des champs feuilletés pour un feuilletages riemanniens sur une variété compacte, C. R. Acad. Sc. Paris 301 (1985), 443–445 Google Scholar

[45] 45.Rawnsley, J. H., f–structures, f–twistor spaces and harmonic maps, in Geometry Seminar Luigo Bianchi II, 1984, Lecture Notes in Mathematics 1164 (Springer-Verlag, 1985), 85–159 Google Scholar | DOI

[46] 46.Sanini, A., Applicazioni armoniche tra i fibrati tangenti di varietá riemanniane, Bol. U.M.I. 6 (2A), (1983), 55–63. Google Scholar

[47] 47.Tondeur, P., Geometry of foliations (Birkhäuser, 1997). Google Scholar | DOI

[48] 48.Wolak, R., Foliated and associated geometric structures on foliated manifolds, Ann. Fac. Sc. Toulouse 10 (1989), 337–360. Google Scholar | DOI

[49] 49.Wolak, R.: The structure tensor of a transverse G-structure on a foliated manifold, Boll. U.M.I. 4-A, (1990), 1–15 Google Scholar

[50] 50.Wolak, R., Geometric structures on foliated manifolds (Santiago de Compostela, 1989). Google Scholar

[51] 51.Wolak, R., Pierrot's theorem for singular Riemannian foliations, Publ. Mat. 38 (1994), 433–439. Google Scholar | DOI

[52] 52.Xin, Y. L., Riemannian submersion and equivariant harmonic maps, in Proceedings of the Symposium an Differential Geometry in honour of S.Buchin (World Scientific Publ., Singapore, 1993), 272–287. Google Scholar

[53] 53.Xin, Y.L., Geometry of harmonic maps (Birkhäuser, 1996). Google Scholar | DOI

[54] 54.Yano, K.: On a structure defined by a tensor field of type (1, 1) satisfying f 3 + f = 0, Tensor, N.S., (1963), 99–109 Google Scholar

[55] 55.Yano, K., Ishihara, S.: On integrability conditions of a structure f satysfying f 3 + f=0, Quart. J. Math., 15 (1964), 217–222 Google Scholar | DOI

Cité par Sources :